The meeting problem in the quantum random walk

TL;DR

This paper investigates the meeting problem for two quantum particles performing a random walk, deriving formulas and analyzing how quantum properties affect the probability of their meeting compared to classical cases.

Contribution

It provides analytic formulas for quantum meeting probabilities, explores effects of entanglement and particle statistics, and compares quantum and classical behaviors.

Findings

Quantum meeting probability decays faster than classical, but not quadratically.

Entanglement and particle statistics influence meeting probabilities.

Asymptotic behavior of quantum meeting probability is characterized.

Abstract

We study the motion of two non-interacting quantum particles performing a random walk on a line and analyze the probability that the two particles are detected at a particular position after a certain number of steps (meeting problem). The results are compared to the corresponding classical problem and differences are pointed out. Analytic formulas for the meeting probability and its asymptotic behavior are derived. The decay of the meeting probability for distinguishable particles is faster then in the classical case, but not quadratically faster. Entangled initial states and the bosonic or fermionic nature of the walkers are considered.